A note on smoothness and differential bases in positive characteristic

TL;DR

This paper establishes criteria for smoothness of algebras with differential bases and for reduced morphisms in the context of noetherian local rings over fields of characteristic p>0, advancing understanding of smoothness in positive characteristic.

Contribution

It provides new smoothness criteria for algebras with differential bases and reduced morphisms in positive characteristic settings.

Findings

Smoothness criteria for algebras with differential bases in characteristic p>0

Smoothness criteria for reduced morphisms

Application to noetherian local rings

Abstract

Let $u:A\to B$ be a morphism of noetherian local rings. We obtain smoothness criteria for algebras with differential bases, in the case of rings containing a field of characteristic $p>0.$ We also give smoothness criteria for reduced morphisms.